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Properties

Label 2-201-201.29-c2-0-2
Degree $2$
Conductor $201$
Sign $-0.317 - 0.948i$
Analytic cond. $5.47685$
Root an. cond. $2.34026$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + (−2 − 3.46i)4-s + (−1 − 1.73i)7-s + 9·9-s + (6 + 10.3i)12-s + (−11.5 + 19.9i)13-s + (−7.99 + 13.8i)16-s + (−13 + 22.5i)19-s + (3 + 5.19i)21-s + 25·25-s − 27·27-s + (−3.99 + 6.92i)28-s + (6.5 + 11.2i)31-s + (−18 − 31.1i)36-s + (−13 + 22.5i)37-s + ⋯
L(s)  = 1  − 3-s + (−0.5 − 0.866i)4-s + (−0.142 − 0.247i)7-s + 9-s + (0.5 + 0.866i)12-s + (−0.884 + 1.53i)13-s + (−0.499 + 0.866i)16-s + (−0.684 + 1.18i)19-s + (0.142 + 0.247i)21-s + 25-s − 27-s + (−0.142 + 0.247i)28-s + (0.209 + 0.363i)31-s + (−0.5 − 0.866i)36-s + (−0.351 + 0.608i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.317 - 0.948i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.317 - 0.948i$
Analytic conductor: \(5.47685\)
Root analytic conductor: \(2.34026\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1),\ -0.317 - 0.948i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.211435 + 0.293729i\)
\(L(\frac12)\) \(\approx\) \(0.211435 + 0.293729i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
67 \( 1 + (54.5 + 38.9i)T \)
good2 \( 1 + (2 + 3.46i)T^{2} \)
5 \( 1 - 25T^{2} \)
7 \( 1 + (1 + 1.73i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (60.5 - 104. i)T^{2} \)
13 \( 1 + (11.5 - 19.9i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (144.5 + 250. i)T^{2} \)
19 \( 1 + (13 - 22.5i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (264.5 + 458. i)T^{2} \)
29 \( 1 + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-6.5 - 11.2i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (13 - 22.5i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + 61T + 1.84e3T^{2} \)
47 \( 1 + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 2.80e3T^{2} \)
59 \( 1 - 3.48e3T^{2} \)
61 \( 1 + (23.5 - 40.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
71 \( 1 + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (71.5 - 123. i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (65.5 + 113. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 + (83.5 - 144. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.37989785180066436827984156223, −11.53511166745489115658758968433, −10.42346652910354195625259955906, −9.875216597608155323357458796976, −8.732509847723348679328551233261, −7.05300930626831523457829107869, −6.26222463153244787473063895177, −5.06127290966401478386616742807, −4.20568913151062147393486670261, −1.59432626674497962336218884986, 0.23989960814668240457223262795, 2.90356805260228794966888239561, 4.50301043024326037944885782471, 5.40100923865472497282781521346, 6.80832184686532910289846919800, 7.77138383248080587331601140001, 8.942703347217719822379619944812, 10.07237000929381844761074315513, 11.02598778836723771974596784903, 12.12590869541105093339618042409

Graph of the $Z$-function along the critical line